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Statistics 108, Regression Analysis PRACTICE MIDTERM EXAM
Open book and notes. Calculator required. There are 5 problems, with a total of 14 parts. Each part of each problem (a, b, etc) is worth 7 points, except Problem 5a, which is worth 9 points.
1. Problem 1.28 on page 37 of your textbook describes data from 84 medium-sized counties in the US. For each county, X = percentage of adults in the county having at least a high-school diploma, and Y = crime rate (crimes reported per 100,000 residents) last year. Here is R output from fitting a simple linear regression model to the data:
lm(formula = CrimeRate ~ HSDiploma, data = CrimeData)
Residuals:
Min 1Q Median 3Q Max
-5278.3 -1757.5 -210.5 1575.3 6803.3
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 20517.60 3277.64 6.260 1.67e-08 ***
HSDiploma -170.58 41.57 -4.103 9.57e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2356 on 82 degrees of freedom
Multiple R-Squared: 0.1703, Adjusted R-squared: 0.1602
F-statistic: 16.83 on 1 and 82 DF, p-value: 9.571e-05
a. Write the population version of the regression model.
b. Write the estimated (sample) regression function.
c. According to the last US Census, 79.8% of Yolo County adults have a high school diploma. Round this number to 80%, and obtain a point estimate for the crime rate in Yolo County.
d. Interpret the slope in the context of this situation.
Problem 1, continued…
e. A 95% confidence interval for E{Yh} when Xh = 70 is 7702 to 9453. Interpret this interval in words, in the context of this situation.
f. The results indicate that counties with higher percentages of high-school graduates tend to have lower crime rates. Can we conclude from this study that having a high school diploma causes people to be less likely to commit crimes, in other words, that higher high-school graduation rates cause crime to be lower? Explain your answer.
2. Define I to be an n × n identity matrix, and H to be the usual hat matrix. A matrix that plays a useful role in regression inference is (I − H). Show using matrix algebra that (I − H) is idempotent.
3. A company offers a training course for the Math SAT. They give their students a test at the end of the course, graded from 0 to 100. They would like to use that test in the future to predict how well students will score on the Math SAT. They have scores on their test and the Math SAT for a sample of students. Thus, X = score on the company’s test and Y = score on the Math SAT. They plan to use the usual simple linear regression model.
a. Would the intercept have a useful meaning in this example? Explain your answer.
b. One of the company analysts states that the intercept should be fixed at 200, because that’s the lowest the SAT Math score can be. Suppose the intercept is set to 200 for this situation. Write the population model.
c. Write the full and reduced models to test whether or not it makes sense to set the intercept to be 200.
d. Write the sum that is to be minimized to get the least squares regression line, if the model you wrote in Part b is used.
4. What assumption is being examined by looking at a normal probability plot? Be specific.
5. A regression equation is to be fit for predicting Y = resting pulse rate using the predictor variables X1 = number of minutes of exercise per week and X2 = gender, with 1 = male and 0 = female. Here are the X values results for 6 individuals:
|
Exercise/week |
200 |
10 |
420 |
50 |
350 |
140 |
|
Gender |
Male |
Female |
Female |
Male |
Male |
Female |
b. Explain in words what the coefficient attached to X2 represents.